Lattices integration method

Gavezzotti, A., Calculation of lattice energies of organic crystals the PIXEL integration method in comparison with more traditional methods. Z. Kristall. 2005, 220, 499-510. 

During the past few decades, various theoretical models have been developed to explain the physical properties and to find key parameters for the prediction of the system behaviors. Recent technological trends focus toward integration of subsystem models in various scales, which entails examining the nanophysical properties, subsystem size, and scale-specified numerical analysis methods on system level performance. Multi-scale modeling components including quantum mechanical (i.e., density functional theory (DFT) and ab initio simulation), atom-istic/molecular (i.e., Monte Carlo (MC) and molecular dynamics (MD)), mesoscopic (i.e., dissipative particle dynamics (DPD) and lattice Boltzmann method (LBM)), and macroscopic (i.e., LBM, computational... 

An alternative method to obtain accurate values of the spin-lattice relaxation time Ej is provided by the TREPR technique with gated MW irradiation, also called the MW-switched time integration method (MISTI)... 

Structure on local nonrandom mixing effects whose many ramifications we wish to imderstand. While improving the predictive abilities of analytical theories, it is also desirable simultaneously to develop the numerically intensive off-lattice integral equation methods that are capable of describing the properties of polymer systems that contain structured monomers, of course, at the expense of heavy numerical computations. (See, for example, recent numerical studies of polyolefin melts using PRISM theory [33-35].)... 

Statistical mechanics methods such as Cluster Variation Method (CVM) designed for working with lattice statics are based on the assumption that atoms sit on lattice points. We extend the conventional CVM [1] and present a method of taking into account continuous displacement of atoms from their reference lattice points. The basic idea is to treat an atom which is displaced by r from its reference lattice point as a species designated by r. Then the summation over the species in the conventional CVM changes into an integral over r. An example of the 1-D case was done successfully before [2]. The similar treatments have also been done for... 

According to the above method, we rewrite the integration, in the irreducible BZ s ment, in terms of the abscissas, a, b and c, along either the reciprocal lattice primitive vectors or any oAer convenient set as follows... 

While the main driving force in [43, 44] was to avoid direct particle transfers, Escobedo and de Pablo [38] designed a pseudo-NPT method to avoid direct volume fluctuations which may be inefficient for polymeric systems, especially on lattices. Escobedo [45] extended the concept for bubble-point and dew-point calculations in a pseudo-Gibbs method and proposed extensions of the Gibbs-Duhem integration techniques for tracing coexistence lines in multicomponent systems [46]. 

Several interesting topics have been excluded, perhaps somewhat arbitrarily, from the scope of this book. Specifically, we do not discuss analytical theories, mostly based on the integral equation formalism, even though they have contributed importantly to the field. In addition, we do not discuss coarse-grained, and, in particular, lattice and off-lattice approaches. At the opposite end of the wide spectrum of methods, we do not deal with purely quantum mechanical systems consisting of a small number of atoms. 

Producing a reasonably good accuracy for analytically defined surfaces, this scheme of calculation is very inaccurate when the field is specified by the discrete set of values (the lattice scalar field). The surface in this case is located between the lattice sites of different signs. The first, second, and mixed derivatives can be evaluated numerically by using some finite difference schemes, which normally results in poor accuracy for discrete lattices. In addition, the triangulation of the surface is necessary in order to compute the integral in Eq. (8) or calculate the total surface area S. That makes this method very inefficient on a lattice in comparison to the other methods. 

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